Herman Weyl in 1916 described the Schwarzschild metric
by a single harmonic function with a pole.  Since then, the Einstein
equation with time symmetry can be regarded as an elliptic variational
problem, and I will report on the recent progress in this direction,
including a series of collaborative work with Gilbert Weinstein and Marcus
Khuri.  We will introduce spactimes of dimension four and five with some
rotational symmetries, and discuss the difference in 4 and 5, and some new
geometric consequences..

The Riemann curvature tensor on a Riemannian manifold induces two
kinds of curvature operators: the first kind acting on two-forms and the
second kind acting on (traceless) symmetric two-tensors. The curvature
operator of the second kind recently attracted a lot of attention due to the
resolution of Nishikawa's conjecture by X.Cao-Gursky-Tran and myself. In
this talk, I will survey some recent works on the curvature operator of the
second kind on Riemannian and Kahler manifolds and also mention some
interesting open problems. The newest result, joint with Harry Fluck at
Cornell University, is an investigation of the curvature operator of the
second kind in dimension three and its Ricci flow invariance.

In complex geometry, the Bergman metric plays a very important role as a
canonical metric as a pullback metric of the Fubini-Study metric of complex
projective ambient space. This work is trying to do something really new to
find a whole new approach of studying hyperbolic complex geometry,
especially for a bounded domain in C^n, we replace the infinite dimensional
complex projective ambient space to the collection of probability
distributions defined on a bounded domain. We prove that in this new
framework, the Bergman metric is given as a pullback metric of the
Fisher-Information metric considered in information geometry, and from this,
a new perspective on the contraction property and biholomorphic invariance
of the Bergman metric will be discussed. As an application of this
framework, in the case of bounded hermitian symmetric domains, we will
discuss about the existence of a sequence of i.i.d random variables in which
the covariance matrix converges to a distribution sense with a normal
distribution given by the Bergman metric, and if more time is left, we will
talk about recent progresses on stochastic complex geometry.

The fundamental gap is the difference of the first two eigenvalues of the Laplace operator, which is important both in mathematics and physics and has been extensively studied. For the Dirichlet boundary condition, the log-concavity estimate of the first eigenfunction plays a crucial role, which was established for convex domains in the Euclidean space and the round sphere. Joint with G. Khan, H. Nguyen, and G. Wei, we obtain log-concavity estimates of the first eigenfunction for convex domains in surfaces of positive curvature and consequently establish fundamental gap estimates. In a subsequent work, together with G. Khan and G. Wei, we improve the log-concavity estimates and obtain stronger gap estimates which recover known results on the round sphere.

Abstract:   Motived by the notion of the algebraic hyperbolicity introduced by X. Chen, we introduce the notion of Nevanlinna hyperbolicity for a pair of (X, D), where X is a projective variety and D is an effective Catier divisor on X. This notion links and unifies the Nevanlinna theory, the complex hyperbolicity (Brody and Kobayashi hyperbolicity), the big Picard type extension theorem (more generally the Borel hyperbolicity). It also implies the algebraic hyperbolicity. The key is to use the Nevanlinna theory on parabolic Riemann surfaces recently developed by Paun and Sibony. This is a joint work with Yan He.